Keller-Segel system, chemotaxis and applications to a mathematical model for Alzheimer's disease

In questa tesi viene presentato il modello di Keller-Segel per la chemiotassi, un sistema di tipo parabolico-ellittico che appare nella descrizione di molti fenomeni in ambito biologico e medico. Viene mostrata l'esistenza globale della soluzione debole del modello, per dati iniziali sufficientemente piccoli in dimensione N>2. La scelta di dati iniziali abbastanza grandi invece può causare il blow-up della soluzione e viene mostrato sotto quali condizioni questo si verifica. Infine il modello della chemiotassi è stato applicato per descrivere una fase della malattia di Alzheimer ed è stata effettuata un'analisi di stabilità del sistema.

A mathematical model of the early visual system and border completion via subriemannian Hamiltonian geodesics

In this thesis, we aim to discuss a simple mathematical model for the edge detection mechanism and the boundary completion problem in the human brain in a differential geometry framework. We describe the columnar structure of the primary visual cortex as the fiber bundle R2 × S1, the orientation bundle, and by introducing a first vector field on it, explain the edge detection process. Edges are detected through a lift from the domain in R2 into the manifold R2 × S1 and are horizontal to a completely non-integrable distribution. Therefore, we can construct a subriemannian structure on the manifold R2 × S1, through which we retrieve perceived smooth contours as subriemannian geodesics, solutions to Hamilton’s equations. To do so, in the first chapter, we illustrate the functioning of the most fundamental structures of the early visual system in the brain, from the retina to the primary visual cortex. We proceed with introducing the necessary concepts of differential and subriemannian geometry in chapters two and three. We finally implement our model in chapter four, where we conclude, comparing our results with the experimental findings of Heyes, Fields, and Hess on the existence of an association field.

Phage therapy: A software system for phage quantification and kinetic model inference.

One of the most serious problems of the modern medicine is the growing emergence of antibiotic resistance among pathogenic bacteria. In this circumstance, different and innovative approaches for treating infections caused by multidrug-resistant bacteria are imperatively required. Bacteriophage Therapy is one among the fascinating approaches to be taken into account. This consists of the use of bacteriophages, viruses that infect bacteria, in order to defeat specific bacterial pathogens. Phage therapy is not an innovative idea, indeed, it was widely used around the world in the 1930s and 1940s, in order to treat various infection diseases, and it is still used in Eastern Europe and the former Soviet Union. Nevertheless, Western scientists mostly lost interest in further use and study of phage therapy and abandoned it after the discovery and the spread of antibiotics. The advancement of scientific knowledge of the last years, together with the encouraging results from recent animal studies using phages to treat bacterial infections, and above all the urgent need for novel and effective antimicrobials, have given a prompt for additional rigorous researches in this field. In particular, in the laboratory of synthetic biology of the department of Life Sciences at the University of Warwick, a novel approach was adopted, starting from the original concept of phage therapy, in order to study a concrete alternative to antibiotics. The innovative idea of the project consists in the development of experimental methodologies, which allow to engineer a programmable synthetic phage system using a combination of directed evolution, automation and microfluidics. The main aim is to make “the therapeutics of tomorrow individualized, specific, and self-regulated” (Jaramillo, 2015). In this context, one of the most important key points is the Bacteriophage Quantification. Therefore, in this research work, a mathematical model describing complex dynamics occurring in biological systems involving continuous growth of bacteriophages, modulated by the performance of the host organisms, was implemented as algorithms into a working software using MATLAB. The developed program is able to predict different unknown concentrations of phages much faster than the classical overnight Plaque Assay. What is more, it gives a meaning and an explanation to the obtained data, making inference about the parameter set of the model, that are representative of the bacteriophage-host interaction.

Modeling and Model Predictive Control of an Industrial Hydraulic System

This thesis aims to illustrate the construction of a mathematical model of a hydraulic system, oriented to the design of a model predictive control (MPC) algorithm. The modeling procedure starts with the basic formulation of a piston-servovalve system. The latter is a complex non linear system with some unknown and not measurable effects that constitute a challenging problem for the modeling procedure. The first level of approximation for system parameters is obtained basing on datasheet informations, provided workbench tests and other data from the company. Then, to validate and refine the model, open-loop simulations have been made for data matching with the characteristics obtained from real acquisitions. The final developed set of ODEs captures all the main peculiarities of the system despite some characteristics due to highly varying and unknown hydraulic effects, like the unmodeled resistive elements of the pipes. After an accurate analysis, since the model presents many internal complexities, a simplified version is presented. The latter is used to linearize and discretize correctly the non linear model. Basing on that, a MPC algorithm for reference tracking with linear constraints is implemented. The results obtained show the potential of MPC in this kind of industrial applications, thus a high quality tracking performances while satisfying state and input constraints. The increased robustness and flexibility are evident with respect to the standard control techniques, such as PID controllers, adopted for these systems. The simulations for model validation and the controlled system have been carried out in a Python code environment.